In The Hobbit: An Unexpected Journey, when Gandalf is walking alongside the Elven king Thranduil, I noticed some similarities between the two, like their height and their hair.
Gandalf seems too naive to expect honest help from the Elven king in such a spooky neighborhood and don't seem to share Thorin's concerns. Could Gandalf be an elf?.
No, Gandalf is not an Elf. While Elves are the Firstborn, the first creatures to be created on Arda, the world of Middle Earth, Gandalf is one of the Maiar, beings somewhat comparable to angels, that were the servants of the Valar, the sort-of gods in the Middle Earth cosmology. The Valar created the world, while the Maiar were their helpers and assistants, and in some cases heralds, standard bearers, messengers or captains.
There is a reason they seem similar, though. The Maiar are spirits, not locked into a single mortal body. When Gandalf, who was originally known as Olórin, went into Middle Earth, he chose a form that is similar to that of the Elves, to fit among them.
Here is how he is described in the Valaquenta, the second part of the Silmarillion, wherein the Valar and Maiar are described:
Wisest of the Maiar was Olórin. He too dwelt in Lórien, but his ways took him often to the house of Nienna, and of her he learned pity and patience [...] for though he loved the Elves, he walked among them unseen, or in form as one of them.
@Avner Shahar-Kashtan's answer is good as far as it goes, but it misses, I think, the key point in the Canon. In the appendices to LotR, JRRT writes about the Istari (the wizards):
When maybe a thousand years had passed, and the first shadow had fallen on Greenwood the Great, the Istari or Wizards appeared in Middle-earth. It was afterwards said that they came out of the Far West...
They came therefore in the shape of Men. though they were never young and aged only slowly. and they had many powers of mind and hand.
This makes it unambiguous that Gandalf's shape was that of an old man. If Thranduil (or any other character) looked like Gandalf, this is because that character also resembled an old human man.
As others have said, Gandalf is not an Elf, nor was he ever one in various drafts/earlier phases (Gandalf was arguably merely a Man in the process of writing The Hobbit, and Tolkien invented the Istari later, after working on Lord of the Rings).
But! And here's the ironic part: Gandalf's name is taken from a real-world source, "Gandálfr" from the Völuspá, in the Poetic Eddas. This name means "wand-elf" (or "staff-elf") in Old Norse!
Many of the Dwarf names in the Hobbit are so sourced, and in earlier drafts of The Hobbit, Gandalf was the name of the leader of the Dwarves (who we know as Thorin), with the wizard called "Bladorthin". So there was really no special intent to link Gandalf (the wizard) with Elves.
That said, Tolkien later retroactively explained this choice of name as being part of his "translated text" frame, whereby the dialect of the men of the north of Middle-earth was translated in Norse-related terms. And so the Dwarves of Erebor, taking "public" names in the vernacular of their region, took Old Norse names in Tolkien's translation of their fictional Westron dialect original. And, further, the Khuzdul name of Gandalf, "Tharkûn", is also glossed as "staff-man".
And Tolkien leans even further into his originally arbitrary choice:
Mostly he journeyed unwearingly on foot, leaning on a staff; and so he was called among Men of the North Gandalf, “the Elf of the Wand”. For they deemed him (though in error, as has been said) to be of Elven-kind, since he would at times works wonders among them, loving especially the beauty of fire; and yet such marvels he wrought mostly for mirth and delight, and desired not that any should hold him in awe or take his counsels out of fear.
Unfinished Tales, Part 4, Chapter II, "The Istari"
And so, ironically, Tolkien settled on the explanation that Gandalf got his name because Men in the north of Middle-earth thought he was one of the Elves!
